Vertex coloring complete multipartite graphs from random lists of size 2

Let K"s"x"m be the complete multipartite graph with s parts and m vertices in each part. Assign to each vertex v of K"s"x"m a list L(v) of colors, by choosing each list uniformly at random from all 2-subsets of a color set C of size @s(m). In this paper we determine, for all fixed s and growing m, the asymptotic probability of the existence of a proper coloring @f, such that @f(v)@?L(v) for all v@?V(K"s"x"m). We show that this property exhibits a sharp threshold at @s(m)=2(s-1)m.